Diagnosis Theory of Arbitrary Domain

Abstract

This chapter presents a general and exact theory for diagnosis of total force and moment exerted to a generic body moving and deforming in a calorically perfect gas. The total force and moment consist of a longitudinal part due to compressibility and irreversible thermodynamics, and a transverse part due to shearing. The latter alone contains the entire force and moment in incompressible flow but is now modulated by the former. The theory represents a full extension of a unified incompressible diagnosis theory of the same type developed by Wu and coworkers to compressible flow, with Mach number ranging from low-speed to moderate supersonic flows. When combined with rapid developed computational fluid dynamics (CFD), the theory permits quantitative identification of various complex flow structures and processes responsible for the forces and moments, and thereby enables rational optimal configuration design and flow control. This theory is further confirmed by a numerical simulation of circular-cylinder flow in the range of free-stream Mach number $M_\infty$ between 0.2 and 2.0. The drags contributed by longitudinal process (L-drag for short) and transverse process (T-drag for short) of the cylinder vary as $M_\infty$ in different ways, of which the underlying physical mechanisms are analyzed. Moreover, each of L-force and T-force integrands contains a universal factor of local Mach number M. Our preliminary tests suggest that the possibility of finding new similarity rules for each force constituent could be quite promising.

Appendix: Derivative Moment Transformation

For self-contained, the derivative moment transformation (DMT) proposed by Wu et al. [19, pp. 700–701] is repeated here. This transformation has appeared in Chaps. 2 and 3 and will be applied to various cases of this chapter.

The one-dimensional prototype of DMT is the familiar integration by parts:

$$\begin{aligned} \int ^b_af(x) \mathrm {d} x = [xf(x)]^b_a-\int ^b_a x f'(x)\mathrm {d} x, \end{aligned}$$

(4.6.1)

which casts the integral of f(x) to that of the moment of its derivative, $xf'(x)$, plus boundary term. This transformation not only makes the new integrand higher peak with narrower support, but also shows the location of the peak matters for the integral. Several DMT identities for higher-dimensional spaces used in this book are the followings.

Let V be a subset of $\mathbf {R}^n$, where $n=2,3$ is the spatial dimension, having a regular boundary $\partial V$ and ${\varvec{f}}$ a vector field defined in V. The following integral identity is applicable:

$$\begin{aligned} \int _V {\varvec{f}}\mathrm {d} V = \frac{1}{n-1}\int _V {\varvec{x}}\times (\nabla \times {\varvec{f}}) \mathrm {d} V - \frac{1}{n-1} \int _{\partial V} {\varvec{x}}\times ({\varvec{n}}\times {\varvec{f}}) \mathrm {d} S, \end{aligned}$$

(4.6.2)

where ${\varvec{x}}$ is the position vector from an arbitrary fixed origin. For a vector field $\phi {\varvec{n}}$, the following identity is also applicable:

$$\begin{aligned} \int _S \phi {\varvec{n}}\mathrm {d} S = -\frac{1}{n-1} \int _S {\varvec{x}}\times ({\varvec{n}}\times \nabla \phi ) \mathrm {d} S + \frac{1}{n-1} \oint _{\partial S} \phi {\varvec{x}}\times \mathrm {d} {\varvec{x}}, \end{aligned}$$

(4.6.3)

where S is a hypersurface of $\mathbf {R}^n$ and $\partial S$ is its boundary. In particular, for $n=3$ there is

$$\begin{aligned} \int _V {\varvec{f}}\mathrm {d} V = - \int _V {\varvec{x}}(\nabla \cdot {\varvec{f}}) \mathrm {d} V + \int _{\partial V} {\varvec{x}}({\varvec{n}}\cdot {\varvec{f}}) \mathrm {d} S, \end{aligned}$$

(4.6.4)

and

$$\begin{aligned} \int _S {\varvec{n}}\times {\varvec{f}}\mathrm {d}S = - \int _S {\varvec{x}}\times [ ({\varvec{n}}\times \nabla ) \times {\varvec{f}}] \mathrm {d}S + \int _{\partial S} {\varvec{x}}\times (\mathrm {d} {\varvec{x}}\times {\varvec{f}}). \end{aligned}$$

(4.6.5)